In this book I have two aims. My first is to give a coherent account of a general method in analytic number theory, and to develop that method sufficiently far that it solves problems otherwise beyond reach. The method applies the simplest notions from functional analysis, and has its roots in geometry.

My second aim, bound to the first, and to me of equal interest, is a light discussion of the creation of the method as a raising of the underlying philosophical motivation into consciousness. In particular, this offers a paradigm for the application of the method itself.

I wrote the present work and my memoir: The Correlation of Multiplicative and the Sum of Additive Arithmetic Functions together. To facilitate a bridge between the two works I have elaborated the treatment of approximate functional equations given in Chapters 2 and 3 of the monograph. In particular, I preserve the same notation. For permission to do this I thank both the American Mathematical Society and Cambridge University Press.

The memoir applies the method to a problem not treated in this book. Background details in the construction of the method are omitted. Consideration of the problem to hand remains paramount. A large number of auxiliary results are required.

The present work is quite different in nature. The method itself is the object of study. Essential inequalities are derived in detail.

When I began actively pursuing the application of approximate functional equations to number theory, in the early seventies, results of the Ulam–Hyers type were sparse. Moreover, they did not lend themselves to the problems which I had to hand.

It should be emphasised that the method of the stable dual is not concerned with the approximate functional equations that arise, for example, in the theory of the Riemann zeta function. In that theory approximate functional equations are established for certain given functions, mainly sums of exponentials. In a sense an analytic reciprocity law is derived. In the method of the stable dual an unknown function is assumed to satisfy a weak global constraint, and as far as possible the local nature of the function is then determined.

As applied to number theory the method of the stable dual typically gives rise to a complicated approximate functional equation involving several functions and many variables. The first step is to tease out an approximate equation of a more manageable type. This step depends upon the number theoretic and distributional properties of the objects under consideration. The appropriate notion of stability is then determined by the number theoretic application in view. My aim was usually towards an equation with continuous rather than discrete variables. Although by 1980 I had developed a tolerable technique for treating approximate functional equations arising in the study of arithmetic functions, I felt the need to better understand some of the arguments.

The notion of duality and its action in analytic number theory informs this entire work. Emphasis is given to the interplay between the arithmetic and analytic meaning of inequalities. The following remarks place ideas employed in the present work within a broader framework.

1. Conies. By duality the notion of a point conic gives rise to the notion of a line conic. The members of the line conic comprise the tangents to the point conic. Slightly surrealistically we may regard a conic to be a geometric object, defined from the inside by a point locus, and from the outside by a line envelope.

2. Dual spaces. Let V be a finite dimensional vector space over a field F. The dual of V is the vector space of linear maps of V into F. The space V and its dual, V′, are isomorphic.

To every linear map T: V → W between spaces, there corresponds a dual map T′: W′ → V′. In standard notation, the action f(x) of a function f upon x is written 〈x, f〉. The dual map T′ is defined by (Tx, y′) = 〈x, T′y′〉 where x, y′ denote typical elements of V, W′ respectively.

Let V = Fn, W = Fm. We may identify W′ with the set of maps W → F given by k ↦ kty′, where y′ is a vector in W, t denotes transposition.